Quantifying tail dependence is an important issue in insurance and risk management. The prevalent tail dependence coefficient (TDC), however, is known to underestimate the degree of tail dependence and it does not capture non-exchangeable tail dependence since it evaluates the limiting tail probability only along the main diagonal. To overcome these issues, two novel tail dependence measures called the maximal tail concordance measure (MTCM) and the average tail concordance measure (ATCM) are proposed. Both measures are constructed based on tail copulas and possess clear probabilistic interpretations in that the MTCM evaluates the largest limiting probability among all comparable rectangles in the tail, and the ATCM is a normalized average of these limiting probabilities. In contrast to the TDC, the proposed measures can capture non-exchangeable tail dependence. Analytical forms of the proposed measures are also derived for various copulas. A real data analysis reveals striking tail dependence and tail non-exchangeability of the return series of stock indices, particularly in periods of financial distress.

The quantification of operational risk has to deal with various concerns regarding data, much more than other types of risk which banks and insurers are obliged to manage. One of the main questions that worries both researchers and practitioners is the bias in the data on the operational losses amounts recorded. We support the assertions made by several authors and defend that this concern is serious when modeling operational losses data and, typically, is presented in all the databases. We show that it's possible, based on mild assumptions on the internal procedures put in place to manage operational losses, to make parametric inference using loss data statistics, that is, to estimate the parameters for the losses amounts, taking in consideration the bias that, not being considered, generates a two fold error in the estimators for the mean loss amount and the total loss amount, the former being overvalued and the last undervalued. In this paper, we do not consider the existence of a threshold for which, all losses above, are reported and available for analysis and estimation procedures. In this sense, we follow a different approach to the parametric inference. Here, we consider that the probability that a loss is reported and ends up recorded for analysis, increases with the size of the loss, what causes the bias in the database but, at the same time, we do not consider the existence of a threshold, above which, all losses are recorded. Hence, no loss has probability one of being recorded, in what we defend is a realist framework. We deduce the general formulae, present simulations for common theoretical distributions used to model (operational reported) losses amounts, estimate the impact for not considering the bias factor when estimating the value at risk and estimate the true total operational losses the bank incurred.

Although unnecessary assumptions are something we all try to avoid, advice on how to do so is much harder to come by than admonition. The most widely quoted dictum on the subject, often referred to by writers on philosophy as “Ockham's razor” and attributed generally to William of Ockham, states “Entia non sunt multiplicanda praeter necessitatem”. (Entities are not to be multiplied without necessity.) As pointed out in reference [I], however, the authenticity of this attribution is questionable.

The same reference mentions Newton's essentially similar statement in his Principia Mathematica of 1726. Hume [3] is credited by Tribus [2c] with pointing out in 1740 that the problem of statistical inference is to find an assignment of probabilities that “uses the available information and leaves the mind unbiased with respect to what is not known.” The difficulty is that often our data are incomplete and we do not know how to create an intelligible interpretation without filling in some gaps. Assumptions, like sin, are much more easily condemned than avoided.

In the author's opinion, important results have been achieved in recent years toward solving the problem of how best to utilize data that might heretofore have been regarded as inadequate. The approach taken and the relevance of this work to certain actuarial problems will now be discussed.

Bias and Prejudice

One type of unnecessary assumption lies in the supposition that a given estimator is unbiased when in fact it has a bias. We need not discuss this aspect of our subject at length here since what we might consider the scalar case of the general problem is well covered in textbooks and papers on sampling theory. Suffice it to say that an estimator is said to be biased if its expected value differs by an incalculable degree from the quantity being estimated. Such differences can arise either through faulty procedures of data collection or through use of biased mathematical formulas. It should be realized that biased formulas and procedures are not necessarily improper when their variance, when added to the bias, is sufficiently small as to yield a mean square error lower than the variance of an alternative, unbiased estimator.

We establish a “top-down” approximation scheme to approximate loss distributions of reinsurance products and Insurance-Linked Securities based on three input parameters, namely the Attachment Probability, Expected Loss and Exhaustion Probability. Our method is rigorously derived by utilizing a classical result from Extreme-Value Theory, the Pickands–Balkema–de Haan theorem. The robustness of the scheme is demonstrated by proving sharp error-bounds for the approximated curves with respect to the supremum and L2 norms. The practical implications of our findings are examined by applying it to Industry Loss Warranties: the method performs very accurately for each transaction. Our approach can be used in a variety of applications such as vendor model blending, portfolio optimization and premium calculation.

I take great pleasure in addressing this audience. As you might know I'm a mathematician with a deep interest in insurance mathematics. As such, it is my sincere opinion that the gap between practising actuaries and theoretical researchers can be made substantially smaller. If my contribution can help in bridging the gap, I will feel fully compensated for the effort it took to prepare this lecture and the results contained therein.

The simple fact that we meet on the occasion of the sixteenth ASTIN Colloquium gives me a challenging opportunity to help in creating a platform on which both theoreticians and practitioners can meet.

The subject of my lecture stems from a long interest in large claims: What are they? Are they really dangerous? Is there a way to get them under control? Can one recognize them in practical situations?

I like to express in simple mathematical terms some results that might help in acquiring better insight on the impact of large claims in insurance mathematics. Perhaps, no result will be of immediate applicability as reality is too complicated to be described by the simplicity of the results to follow. Nevertheless the latter can be considered as building blocks of a real world in which one has to tackle large claims in theory as well as in practice.

The purpose of this paper is to show how linear programming methodology can help us to design Bonus-Malus premium scales with some interesting theoretical and practical attributes. Examples of these properties are the financial equilibrium of the system, the monotonicity and proper variability of the premium scale, and the improvement of some efficiency measures such as the RSAL and the elasticity of the system. We will conclude that the use of the linear programming methodology makes possible a high degree of interaction between the designer and the mathematical model.

A generalization of range-value-at-risk (RVaR) and tail-value-at-risk (TVaR) for d-dimensional distribution functions is introduced. Properties of these new risk measures are studied and illustrated. We provide special cases, applications and a comparison with traditional univariate and multivariate versions of the TVaR and RVaR.

1. The claims experience of a motorcar insurance is assumed to give some indication about the risk (basic claim frequency) of that insurance. The experience rating systems in motor insurance are based on this supposition. In these systems the premium to be paid in some year is a function of the individual claims experience of preceding years. That premium can be considered as the difference between:

The bonus usually consists of at least the following components:

a. a component concerning the individual claim frequency.

b. an individual random factor.

c. a collective random factor.

Other factors, like the effect of the trend in the claim frequencies and the effect of the dependence of the claim frequency of age and experience of the driver are not taken into account in this paper.

2. The meaning of the three components mentioned in section I can be demonstrated by the following example.

The structure function U(χ) of the portfolio, being the distribution function of the individual basic claim frequencies χ is assumed to be defined by representing a gamma distribution with expected value q and variance qb. The probability Pi for an insurance to have i claims in a certain year is supposed to be

In this formula the factor y has the same value for all insurances in the portfolio in a certain year but varies from year to year according to the distribution function G(y) with expected value 1 and variance .

The paper extends earlier results by demonstrating that there is an optimal range of values for the period for amortizing valuation surpluses or deficiencies, in the case when there is a one year time delay between fixing a contribution rate and the accounting information about current fund levels. The optimal range is compared for the cases where there is no time delay and there is a one year time delay.

In this paper, we aim to evaluate the distribution of the aggregate claims in the collective risk model. The claim count distribution is firstly assumed to belong to a generalised (a, b, 0) family. A matrix form recursive formula is then derived to evaluate the related compound distribution when individual claim amounts follow a discrete distribution on non-negative integers. The corresponding formula is also given for continuous individual claim amounts. Secondly, we pay particular attention to the recursive formula for compound phase-type distributions, since only certain types of discrete phase-type distributions belong to the generalised (a, b, 0) family. Similar recursive formulae are obtained for discrete and continuous individual claim amount distributions. Finally, numerical examples are presented for three counting distributions.

In recent years, the determination of premium principle under various non-expected utility frameworks has become popular, such as the pioneer works by Tsanakas and Desli (2003) and Kaluszka and Krzeszowiec (2012). We here revisit the problem under another prevalent behavioral economic theory, namely the Disappointment Aversion (DA) Theory proposed by Gul (1991). In this article, we define and study the properties of the DA premium principle, which builds on the equivalent utility premium principle. We derive various properties of this premium principle, such as non-negative and no unjustified risk loading, translation invariance, monotonicity, convexity, positive (non-)homogeneity, independent (non-)additivity, comonotonic (non-)additivity and monotonicity with respect to the extent of disappointment. A generalized Arrow–Pratt approximation is also established. Explicit representations of the premium principle are obtained for linear and exponential utilities, and they reveal that the premium principle proposed echoes the capital reserve regulatory requirement in practice.

In this paper we will show how to set up a practical bonus-malus system with a finite number of classes. We will use the actual claim amount and claims frequency distributions in order to predict the future observed claims frequency when the new bonus-malus system will be in use. The future observed claims frequency is used to set up an optimal bonus-malus system as well as the transient and stationary distributions of the drivers in the new bonus-malus system. When the number of classes as well as the transition rules of the new bonus-malus system have been adopted, the premium levels are obtained by minimizing a certain distance between the levels of the practical bonus-malus system and the corresponding optimal bonus-malus system. Some iterations are necessary in order to reach stabilization of the future observed claims frequency and the levels of the practical bonus-malus system.

This paper studies dynamic reinsurance contracting and competition problems under model ambiguity in a reinsurance market with one primary insurer and n reinsurers, who apply the variance premium principle and who are distinguished by their levels of ambiguity aversion. The insurer negotiates reinsurance policies with all reinsurers simultaneously, which leads to a reinsurance tree structure with full competition among the reinsurers. We model the reinsurance contracting problems between the insurer and reinsurers by Stackelberg differential games and the competition among the reinsurers by a non-cooperative Nash game. We derive equilibrium strategies in semi-closed form for all the companies, whose objective is to maximize their expected surpluses penalized by a squared-error divergence term that measures their ambiguity. We find that, in equilibrium, the insurer purchases a positive amount of proportional reinsurance from each reinsurer. We further show that the insurer always prefers the tree structure to the chain structure, in which the risk of the insurer is shared sequentially among all reinsurers.

In this study, we propose new risk measures from a regulator’s perspective on the regulatory capital requirements. The proposed risk measures possess many desired properties, including monotonicity, translation-invariance, positive homogeneity, subadditivity, nonnegative loading, and stop-loss order preserving. The new risk measures not only generalize the existing, well-known risk measures in the literature, including the Dutch, tail value-at-risk (TVaR), and expectile measures, but also provide new approaches to generate feasible and practical coherent risk measures. As examples of the new risk measures, TVaR-type generalized expectiles are investigated in detail. In particular, we present the dual and Kusuoka representations of the TVaR-type generalized expectiles and discuss their robustness with respect to the Wasserstein distance.

Ce rapport n'a pas le but de résumer toute la littérature actuarielle sur les assurances non viagères: on se proposait de tracer le fil conducteur de quelques uns des développements d'idées où l'apport des mathématiques est plus profond et, en particulier, de ceux où il semble qu'il y aura encore à travailler longtemps. A ce compte, seulement quelques étapes pouvaient être explicitement mentionnées dans le texte, tandis qu'une bibliographic complète aurait dû suivre à la fin du rapport.

Maintes circonstances ont restreint le temps à disposition des rapporteurs et ont augmenté les difficultés d'un travail en commun à distance; sans doute quelques contributions ont échappé ou n'ont pas reçu le relief qu'il aurait fallu leur donner. Les auteurs prient de bien vouloir justifier les omissions par ces circonstances, et de les leur signaler.

L'assurance sur la vie et les assurances non viagères semblent présenter des différences essentielles, soit au point de vue de leur objet (indemnité d'une perte dans le deuxième cas, mais non dans le premier), soit au point de vue de leur méthode (théorie mathématique complétement développée dès longtemps pour le premier cas, mais non dans le deuxième).

Ya-t-il néanmoins une notion unique d'assurance, en général, qui comprenne toutes les deux? Est-il possible de concevoir d'une façon unique l'application du calcul des probabilités et des méthodes mathématiques qui le traduisent, dans le champ de l'assurance en général? Ou ya-t-il des limitations, ou des circonstances qui obligent à distinguer à cet effet les deux champs de l'assurance sur la vie et de l'assurance non viagère?

A functional approach is taken for the total claim amount distribution for the individual risk model. Various commonly used approximations for this distribution are considered, including the compound Poisson approximation, the compound binomial approximation, the compound negative binomial approximation and the normal approximation. These are shown to arise as zeroth order approximations in the functional set-up. By taking the derivative of the functional that maps the individual claim distributions onto the total claim amount distribution, new first order approximation formulae are obtained as refinements to the existing approximations. For particular choices of input, these new approximations are simple to calculate. Numerical examples, including the well-known Gerber portfolio, are considered. Corresponding approximations for stop-loss premiums are given.

The Ministry of Social Affairs and Health, being the Supervising Office of Insurance in Finland, has established a special working group to investigate the problems involved with the solvency of insurers. A report will be compiled in a near future. The capacity of risk carriers is one of the problems dealt with, and it will be preliminarily reviewed in this paper.

The problem was treated by the working group parallelly by means of

1. an empirical approach observing actual fluctuations in underwriting gains of insurers, and

2. a theoretical approach, constructing a stochastic-dynamic model and studying its behaviour, especially its sensitivity to numerous background factors.

First the methods of investigation are described and their application is then demonstrated using some numerical data. Because a comprehensive report will be published by the working group separately, only the main schedule is given. For the same reason the consideration is limited here to stochastic risks, omitting the fact that the solvency of an insurer is also jeopardized by numerous “non-stochastic” risks such as failure in investments, political interference of the authorities, mismanagement of the company, or misappropriation of its property.

A retrospective rating plan, whose insurance premium depends upon an insured's actual loss during the policy period, is a special insurance agreement widely used in liability insurance. In this paper, the design of an optimal retrospective rating plan is analyzed from the perspective of the insured who seeks to minimize its risk exposure in the sense of convex order. In order to reduce the moral hazard, we assume that both the insured and the insurer are obligated to pay more for a larger realization of the loss. Under the further assumptions that the minimum premium is zero, the maximum premium is proportional to the expected indemnity, and the basic premium is the only free parameter in the formula for retrospective premium given by Meyers (2004) and that the basic premium is determined in such a way that the expected retrospective premium equates to the expected indemnity with a positive safety loading, we formally establish the relationship that the insured will suffer more risk for a larger loss conversion factor or a higher maximum premium. These findings suggest that the insured prefers an insurance policy with the expected value premium principle, which is a special retrospective premium principle with zero loss conversion factor. In addition, we show that any admissible retrospective rating plan is dominated by a stop-loss insurance policy. Finally, the optimal retention of a stop-loss insurance is derived numerically under the criterion of minimizing the risk-adjusted value of the insured's liability where the liability valuation is carried out using the cost-of-capital approach based on the conditional value at risk.

Une Compagnie d'assurances délivre par l'intérmediaire d'Agences de Voyage des contrats d'assurances individuelles garantissant les personnes prenant place dans des autocars contre les décès par accident. Le capital garanti est uniforme. Il sera pris égal à I par simplification.

Cette Compagnie prend à sa charge le paiement de i décès au maximum dans chaque accident et demande à un Réassureur de payer, aux termes d'un traité en excédent de sinistre, les autres décès; autrement dit, si un accident cause d décès, la Compagnie paie I, le Réassureur en excess paie d — I.

Il s'agit de calculer la prime de ce traité en excédent de sinistre?

La solution dépend de trois paramètres:

n nombre de passagers par autocar, égal au maximum au nombre de places; on supposera, pour fixer les idées, n = 502).

d nombre de décès survenus dans un même accident. C'est une variable stochastique comprise entre 0 et n.

Il n'y a pas lieu de faire intervenir la fréquence d'accidents (entraînant des morts) q, car celle-ci est un des éléments constitutifs du tarif appliqué par la Cédante et la prime de réassurance est exprimée comme une fraction de la prime originale. Si celle-ci est incorrecte, trop basse ou trop élevée, l'Assureur réalise une perte ou un bénéfice et il est normal que le Réassureur suive en cela la fortune de sa Cédante. La seule question est celle d'un partage équitable de la prime perçue, bonne ou mauvaise, entre la Cédante et le Réassureur.